Paraconsistent logic
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A paraconsistent logic is a logic which attempts to deal with contradictions.
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Motivations
There are several motivations for paraconsistent logic, all of which arise out of a dissatisfaction with the consistency of classical logic.
- The semantic paradoxes, especially those of self-reference, provide formal reasons for questioning classical logic. Consider the liar paradox (where "<L>" stands for "the proposition that L"):
(L) <L> is not true.
Plugging L into itself, we get
"<L> is not true" is not true
which appears to say the same thing as
(L' ) L is true
(This inference is based on several rather plausible, but admittedly not unassailable, premises about double negation elimination and the relationship between <P> and P--that is, the relationship between a proposition and the state of affairs to which the proposition corresponds. Roughly, we call this relation "truth", and so we are able, in a sense, to move in and out of the quotation marks and proposition-marking brackets.) And, if we continue to operate on a certain naive assumption about the nature of truth, L looks like the negation of L' . Thus, a contradiction. (Russell's paradox of set theory and higher-order logics come up against similar problems.)
A staunch subscriber to classical logic (or, more generally, consistent logics) might simply ignore such problems, or say that sentences like L are simply meaningless. Understandably, paraconsistent logicians are wary to accept this line; after all, "This sentence is false" seems a perfectly coherent, if thought-provoking, sentence. Adopting a position according to which grants the truth of a sentence like L, as well as its apparent negation L' , is a possible way out of the semantic paradoxes.
- Less formally, one might think that our actual reasoning is paraconsistent. Graham Priest, proponent of the paraconsistent logic dialetheism, offers the example of a man who is standing in a doorway in such a way that precisely half of his molecules are in the room and half of them not. What might we say about the choice between the man's utterance of "I am currently in the room" as opposed to its negation "I am not currently in the room" (1998)? That we would allow for both to be true is not an altogether outlandish solution.
Problems
In classical logic, if <math>\Lambda \vdash P<math> and <math>\Lambda \vdash \neg P<math>, i.e. there is some theory <math>\Lambda<math> which allows you to show both <math>P<math> and <math>\neg P<math>, then it is possible to prove that every formula <math>A<math> (and so its negation <math>\neg A<math>) is true in the proof calculus; a similar model theoretic result can be derived. This is called the principle of explosion, since the set of theorems explodes (indeed, to include every sentence in the language) when it is instantiated. Classical logic, intuitionistic logic, and indeed most other logics suffer from this problem. Paraconsistent logics must not fall into this trap.
A paraconsistent might, in order to solve this problem, simply reject the principle of explosion. Of course, this is not such a trivial thing to do. Explosion is a direct consequence of our truth-functional notion of disjunction; to reject the former is to bring into question the latter, which is, it seems, well-founded if anything is.
Some paraconsistent logics:
- Dialetheism
- Multi-valued logics may support paraconsistent truth values
- Relevance logic supports a four-valued notion of truth: true, false, neither-true-nor-false and the paraconsistent both-true-and-false.
In Knowledge representation, much attention is paid to reasoning systems that are indefeasible, that is they may support conclusions which they later reject when more complete evidence is available. Arguably, indefeasible logics are paraconsistent.
Paraconsistent logic might be used as a basis for paraconsistent mathematics, allowing inconsistencies without turning all statements into theorems.
Sources
Parsons, Terence. “True Contradictions.” Canadian Journal of Philosophy 20 (1990): 335-354.
Priest, Graham. “What Is So Bad About Contradictions?” Journal of Philosophy 95 (1998): 410-426.
Priest, G., Routley, R., and Norman, J. (eds.) Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, München, 1989.
Priest, G. & Tanaka, K., "Paraconsistent Logic", The Stanford Encyclopedia of Philosophy (Winter 2004 Edition), Edward N. Zalta (ed.) [1] (http://plato.stanford.edu/archives/win2004/entries/logic-paraconsistent/).